0
votes
0answers
15 views

Proving that the function set $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set

I have the the following problem from my Fourier analysis book: Show that $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set in $PC(0,l)$, i.e. class of piecewise ...
4
votes
0answers
62 views

Delicate Integral $I=\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set a=0 we get a similar integral given by $$ ...
0
votes
1answer
22 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
3
votes
1answer
38 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
2
votes
0answers
51 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
1
vote
1answer
64 views

Evaluate $\int_0^{\infty}\frac{x^4e^x}{(e^x-1)^2} \, dx$

I am trying to find the value of the integral below. Can anyone let me know how to evaluate this integral? $$\int_0^{\infty}\frac{x^4e^x}{(e^x-1)^2} dx$$
0
votes
2answers
85 views

Compute $\int_0^\infty \frac{\ln x}{(1+x)^3}\,\mathrm{d}x$

Compute $$\int_0^\infty \frac{\ln x}{(1+x)^3}\,\mathrm{d}x$$ Well by comparison test the integral is convergent. I tried to use residue theorem, with the positive real axis being the branch ...
0
votes
1answer
33 views

Computing ${\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right)$

Let $t \in \mathbb{R}$. Is the following elementary calculation correct? $$ {\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right) = \underbrace{{\mathrm{d} \over \mathrm{d}t}\left(it\right) \cdot ...
1
vote
1answer
47 views

Computing $\int_{|z|=1} {e^z \over z}\ dz$

Goal: Let $\gamma$ be the unit circle. Then I aim to compute $$ \int_{|z|=1} {e^z \over z}\ dz = \int_{\gamma} {e^z \over z}\ dz $$ Attempt: Consider that $\gamma$ is a closed curve. Let $a = 0$. ...
1
vote
2answers
33 views

Showing that ${d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)}$

I'm trying to show that $$ {d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)} $$ However my attempt yields that $$ {d \over dz}\log\left[ z - a \over z - b ...
1
vote
0answers
43 views

Cauchy Integral Theorem problem (lack of understanding)

First of all i was asked to evaluate this integral $\int_\gamma \frac{2z}{(z-1)(z-3)} dz$ where $\gamma (t) = 2e^{it}$ for $0\leq t \leq 2\pi$. Now I thought I would have to calculate this ...
2
votes
1answer
53 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
6
votes
2answers
103 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
0
votes
0answers
18 views

integrating of complex exponential function

I know $\int x^a dx=\frac{x^{a+1}}{a+1}$ when $a$ is real. How I can calculate this integral when $a$ is complex?
0
votes
3answers
49 views

Algebraic Equation?

$$Ve^{i\theta} = We^{i\phi}$$ where, $V$ and $W$ are some real constants. From this my book concludes: $\theta = \phi$. How does it conclude this? I don't see why its valid to just equate the ...
2
votes
1answer
33 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
0
votes
2answers
45 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
0
votes
1answer
69 views

Why isn't $i$ affected by powers?

When finding roots of complex functions we can write for example: $$z=2-2i$$ Let's find complex numbers $w$ such that $$w^4 = 2-2i$$ $$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$ This reads: ...
0
votes
0answers
17 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
1
vote
0answers
18 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
2
votes
1answer
24 views

Examples of vector field that is continuously differentiable but not conservative?

I am just curious what would be the case in which a vector field ($\vec f :\Bbb R^2 \rightarrow \Bbb R^2$) is well-defined and continuously differentiable on a region R enclosed by a simple closed ...
4
votes
0answers
77 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
0
votes
0answers
19 views

How to find the tangential component of Velocity?

Let $v=-\nabla \phi$, where $\phi$ is the velocity potential. I am interested in to find the value of $|v_{tan}|$ which is the tangential component of velocity.
1
vote
1answer
84 views

About asymptotic behaviour of a divergent integral.

I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. For the logarithm I am assuming a branch-cut along the positive imaginary axis starting at $x = ia$. ...
1
vote
1answer
54 views

consequences of Schwarz lemma of holomorphic functions of unit disk

Let $D$ be the open unit disk centered at $0$ in the complex plane. Let $f:D\longrightarrow D$ be holomorphic such that $f(0)=0$. Use the Schwarz lemma to prove that $|f(z)+f(-z)|\leq 2|z|^2$ for any ...
0
votes
2answers
53 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
0
votes
1answer
52 views

proving a limit of a function by definition

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$ Prove that $$ \lim_{z \to i} f(z) = i $$ using the ...
0
votes
1answer
41 views

Complex Fourier series of $f(\theta) = e^{\theta}$

I have the following Fourier series problem: Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let ...
0
votes
0answers
30 views

Openness of Sets - Point set Topology

What's the formal method called where we find a $p>0$ such that $|(x,y)-(x_0,y_0)|$ is less than $p$?
0
votes
1answer
29 views

Complex polynomial decomposition - Residue Theory

I am given the following function: $R(z) = (z^2-9)/(z^2+9)^2 $ I need to let $R = P/Q$ be a rational function with $deg P < deg Q$. I will let $ξ$ be a pole of $R$ and the coefficient of $1/(z-ξ)$ ...
0
votes
0answers
25 views

If $Z$ is an admissible function, does $f(z) = f(|z|)$?

If $Z$ is an admissible function, does $f(z) = f(|z|)$? For example, if $f(z) = x^3 + 1$ and I am given $2$ points $z_1 = -1+i\sqrt3$ and $z_2 = -1-i\sqrt3$, can I just find the moduli and use that ...
1
vote
1answer
66 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let $2$ points $z_1 = (-1+i\sqrt3)/2$ and $z_2 = (-1-i\sqrt3)/2$ I am trying to show that there is no point $w$ on the line segment from $z_1$ to ...
3
votes
2answers
72 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
2
votes
2answers
61 views

Find all singularities of a function

Let $f(z)=z(e^{\frac 1 z} -1)\tan{\frac 1 {z-1}}$. Find all zeros and singularities of $f$. I know that $f$ is analytic in $\{z:z\not=0,1,\frac 1 {\pi k+\frac \pi 2}\}$ where $k\in\mathbb{Z}$ and that ...
1
vote
1answer
39 views

The definition of Residue

In Wikipedia the definition of a residue of a function $f$ in a point $a$ is a unique value $R$ such that $f(z)-\frac{R}{z-a}$ has an anti derivative in a punctured disk $0<|z-a|<\delta$. How is ...
0
votes
1answer
39 views

Analytical Formula for Hilbert Transform of a Ricker Wavelet

I am attempting to validate some numerical code I have to compute Hilbert transforms. As I am interested in the Hilbert transforms of functions with rapid decay, I wanted to unit test my code with the ...
0
votes
1answer
38 views

Why does $\sum_{n\neq0}\:\left|\frac{a_n}{n}\right| \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2}$?

My question is: Why does $$\sum_{n\neq0}\:\left|\frac{a_n}{n}\right| \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2},$$ where $a_n$ is some complex number, $n$ an integer going ...
0
votes
0answers
22 views

Providing an upper bound for a sum of complex numbers

Let $(\alpha_l)_{l=0}^{k-1}$ and $(\beta_l)_{l=0}^{m-1}$ be two sequences of complex numbers where $m>k$. It is known that $$ 0<\frac{1}{m}\sum_{l=0}^{m-1}|\beta_l|^2\leq A $$ where $A>0$. ...
1
vote
2answers
63 views

Contour Integral for Cosine and a rational function

I've been trying to figure out this integral via use of residues: $$\int_{-\infty}^{\infty} \displaystyle \frac{\cos{5x}}{x^4+1}dx$$ The usual semicircle contour wont work for this guy as the ...
3
votes
2answers
59 views

Differentiate complex function?

$$f(z)=3z^2+\bar{z}$$ I want to show the function is either differentiable or not so I can state if it is holomorphic or not. What is the method for this ? Edit - Can some give an example of how to ...
0
votes
3answers
35 views

About the Scalar product

These are the lecture notes of my teacher and I am getting confused how he reached at $V_1$.$V_2$= Re($z_1$$z_2$). Can anyone help me to understand this.
3
votes
2answers
78 views

Elimination of Trigonometric Functions

Is there a simple way to eliminate the trigonometric functions here? $$ \begin{array}{lcl} A\cos(3\omega\tau)+B\sin(3\omega\tau)+C\cos(\omega\tau) &=& D\\ ...
3
votes
2answers
109 views

Trouble with $\int_0^\infty e^{-ix^2}\mathrm{d}x$

I'm trying to evaluate $$ \int_0^\infty \mathrm{d}x\ e^{-ix^2}. $$ I tried to integrate on the following contour $\Gamma_R$: the frontier of a circular sector, centered at the origin, of angle $\pi / ...
3
votes
2answers
64 views

Justifying an ODE's solution

In an introductory lesson into ODEs, in order to "semi-rigorously" justify the solution for e.g. : $(a)\ \ y'+y=0$ we proceed without an ansatz or guess solution (hence the "semi-rigour"): Let: ...
3
votes
2answers
76 views

Asymptotics of coefficients

This is a question that asks the reader for a $strategy$ to solve a particular problem. I cannot solve this problem myself so I am looking around for general methods one might use to confront it with. ...
1
vote
1answer
25 views

Show that the composition of the two functions is the identity.

I have to check that the composition of the following functions gives the identity (or that one function is the inverse of the other): $$\pi:S^2\backslash \{N\}\to\mathbb{C}$$ $$(x_1,x_2,x_3) ...
0
votes
2answers
92 views

Prove that $f(z) = \sum\limits_{k = 1}^\infty \frac{z^{2^k}}{2^k}$ is continuous in the closed unit disc and holomorphic inside it.

I have started off by assuming that there is a disc of radius $r$ for which $|z|<r$ for $r \in (0,1)$ and $z \in D_r$. This implies that $|z|^{2^k} < r^{2^k}$ And after that, I don't know ...
4
votes
1answer
144 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
2
votes
1answer
38 views

Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
3
votes
1answer
32 views

number of zeros in a disk of a holomorphic function

Let $f$ be a holomorphic function defined in a beighborhood of $\overline{D(0,R)}$ which has no zeros on $\partial D(0,R)$. Let $N$ be the number of zeros of $f$ inside $D(0,R)$. Prove that ...